Breaking the fundamental scattering limit with gain metasurfaces

A long-held tenet in physics asserts that particles interacting with light suffer from a fundamental limit to their scattering cross section, referred to as the single-channel scattering limit. This notion, appearing in all one, two, and three dimensions, severely limits the interaction strength between all types of passive resonators and photonic environments and thus constrains a plethora of applications in bioimaging, sensing, and photovoltaics. Here, we propose a route to overcome this limit by exploiting gain media. We show that when an excited resonance is critically coupled to the desired scattering channel, an arbitrarily large scattering cross section can be achieved in principle. From a transient analysis, we explain the formation and relaxation of this phenomenon and compare it with the degeneracy-induced multi-channel superscattering, whose temporal behaviors have been usually overlooked. To experimentally test our predictions, we design a two-dimensional resonator encircled by gain metasurfaces incorporating negative- resistance components and demonstrate that the scattering cross section exceeds the single- channel limit by more than 40-fold. Our findings verify the possibility of stronger scattering beyond the fundamental scattering limit and herald a novel class of light-matter interactions enabled by gain metasurfaces.

In this manuscript, the authors proposed novel scattering enhanced method that overcomes the singlechannel scattering limit by exploiting gain media. As a demonstration, the authors have design a twodimensional resonator with tunnel diode in order to provide a negative differential resistance. Overall, the authors have presented an interesting concept to exploit gain metasurfaces.
The manuscript is well-written and structured. It shows studies on the dynamic of these media and it shows a progress in the scattering enhancement limit with respect previous achievements. However, some points must be addressed for publication in Nature Communications.
1-The authors claim that the total scattering cross section is more than 40 times the single-channel scattering at 3.55GHz, however no experimental point at the maximum enhancement is provided. Experimental point in this frequency must be provided in order to demonstrate the claimed enhancement.
2-An experimental measurement of the dielectric rod without gain metasurface could be very helpful to discard border effects in the electromagnetic set up measurements.
3-Taking into account that the experimental design and the results are based in the numerical simulation of CST studio commercial software, the numerical method must be more detailed (boundary conditions, mesh, how the metallic materials are considered, etc.) Reviewer #2 (Remarks to the Author): The manuscript entitled «Breaking the fundamental scattering limit with gain metasurfaces » by Chao Qian, Yi Yang, Yifei Hua, Chan Wang, Xiao Lin, Tong Cai, Dexin Ye, Erping Li, Ido Kaminer, and Hongsheng Chen has been reviewed.
The manuscript presents a new and elegant way to achieve superscattering without relying on embedded particles in extreme environment or relying on multiple resonances. Here the authors propose to utilize gain at the nanoparticle position to assist superscattering behavior. The difference with existing approaches is that superscattering can occur with a single resonance by compensating the radiation leakage relying on spatial and temporal gain modulation. A reshaping of the radiation patterns in the superscattering regime is also observed.
I found the idea quite innovative and certainly worth publication.
I have a few comments: The discussion on the steady state cross section affecting the transient response is quite paradoxal, at least to me. For device with extremely large cross section, I would expect extremely fast relaxation but instead the numerical results indicate the opposite. It might be worth elaborating on this unintuitive observation. Is it because of the radiation leakage compensation by amplification that artificially increases the mode lifetime? Reciprocally, the energy accumulation takes longer as well.
The manuscript reports about 40 time the scattering cross section in the active regime. It would be interesting to show how this number scale as a function of relevant parameter such as gain/loss ratio.
I found strange links in pdf, such as line 137, that directly point to Wikipedia…

Response Letter to Reviewers
We are grateful for the constructive comments on this manuscript (NCOMMS-22-12290) from all the referees.
In the text below, each comment is quoted in italics and is followed by the corresponding detailed response.
We have also revised the manuscript and supplementary material accordingly. These updates are highlighted in blue and by a vertical red line in the left margin in those files. In the text below, the references to these updates are highlighted in a similar way (i.e., by a vertical red line).

General comments from Referee #1:
In this manuscript, the authors proposed novel scattering enhanced method that overcomes the single-channel scattering limit by exploiting gain media. As a demonstration, the authors have design a two-dimensional resonator with tunnel diode in order to provide a negative differential resistance. Overall, the authors have presented an interesting concept to exploit gain metasurfaces.
The manuscript is well-written and structured. It shows studies on the dynamic of these media and it shows a progress in the scattering enhancement limit with respect previous achievements. However, some points must be addressed for publication in Nature Communications.

Authors Response:
We thank the referee for his/her positive comments. In the following, we address the specific comments pointby-point whilst revising our manuscript.

Specific comments from Referee #1:
Referee #1 --Comment 1: 1-The authors claim that the total scattering cross section is more than 40 times the single-channel scattering at 3.55GHz, however no experimental point at the maximum enhancement is provided. Experimental point in this frequency must be provided in order to demonstrate the claimed enhancement.

Authors Response:
We thank the referee for pointing this out. As suggested, we have conducted a new experiment to demonstrate the scattering enhancement at 3.55 GHz. Figure R1 shows the near-field distribution and radar cross section . From Fig. R1, the measured near-field distribution is consistent with the simulated case. To quantitatively characterize the scattering performance, we calculated the total scattering cross section via = 1 2 ∫ 2 0 . The calculation result turns out that is 41.9 times the single-channel scattering limit.
In the new submission, we have replaced the middle panel of Fig. 4d by Fig. R1 and added the description on lines 194-195 of the main text.
"Experimental result indicates that the total scattering cross section is more than 40 times the single-channel scattering limit at 3.55 GHz."

Authors Response:
This is an important question. In the new version, we have added the experimental evidence for the dielectric rod without gain metasurface in Fig. R2. Compared with the dielectric rod with gain metasurfaces (Fig. 4 in the main text and Fig. R1), the wavefront is flat and the scattering is almost negligible for pure dielectric rod. In addition, the is very low (Fig. R2b), in a good agreement with the simulated case.
In the new submission, we have added the above discussion in the supplementary materials and referred to it on lines 237-238.
"Before the experiment, we measured the near-field for the pure dielectric rod without gain metasurfaces to check the border effects ( Supplementary Fig. 12)." Figure R2 | Experimental result of the pure dielectric rod without the gain metasurface. For conceptual demonstration, the measured frequency is 3.55 GHz. a, Near-field simulation and measurement of the total field in a region close to the dielectric rod. b, Simulated and measured radar cross sections. Notice that the scale in b is much smaller than that in Fig. R1.

3-Taking into account that the experimental design and the results are based in the numerical simulation of
CST studio commercial software, the numerical method must be more detailed (boundary conditions, mesh, how the metallic materials are considered, etc.)

Authors Response:
Thanks for the careful reading. In the new submission, we have added the simulation details in Methods section; see below.
"The scattering parameters of the gain metasurfaces are simulated using CST Microwave studio. The gain metasurface comprises copper patch and substrate layer with a relative permittivity of 3.4 and a thickness of 0.05 mm. A tunnel diode is incorporated into each metasurface element, whose equivalent circuits are embedded in the simulation. We applied the EM-circuit co-simulation to connect the metasurface and tunnel diode. The "FSS, Metamaterial-Unit Cell template" in the "Frequency domain solver" is chosen, with unit cell boundary conditions enabled in the x-and z-directions and open boundary conditions in the y-direction (Fig.   3a). Finally, an adaptively refined tetrahedral mesh was applied to the simulated structures with a number of about 35,000 cells. The maximum mesh size was set at approximately 4.8 mm, which is about one-eighteenth of a free-space wavelength at 3.5 GHz."

General comments from Referee #2:
The manuscript entitled « I found the idea quite innovative and certainly worth publication.

Authors Response:
We are grateful to the referee for these positive comments and acknowledging that "I found the idea quite innovative and certainly worth publication". In the following, we address the specific comments point-by-point and revise our manuscript correspondingly.

Specific comments from Referee #2:
Referee #2 --Comment 1: I have a few comments: Reciprocally, the energy accumulation takes longer as well.

Authors Response:
We thank the referee for these very constructive suggestions. The numerical results are actually consistent with our expectation-a larger scattering cross section requires longer time for accumulating and releasing energy; this agrees with the study [Phys. Rev. Lett. 106, 165503 (2011)].
According to the above results, we found that an object with larger scattering cross section corresponds to longer buildup time (∆ 1 > ∆ 2 > ∆ 3 ). This is understandable because the larger the total scattering cross section, the longer the time required to accumulate sufficient energy to reach a stable state. Reciprocally, the relaxation time is longer. In contrast, if an object has a tiny scattering cross section, the buildup/relaxation time will be very short, as exemplified in Fig. R3c. For a gain system (Fig. R3a), when the incident wave cuts off abruptly, the field scattered by the single-channel superscatterer decays slowly. This exotic scattering phenomenon occurs not only because it has accumulated much scattering energy in the buildup period, but also the radiation leakage is compensated by the gain metasurfaces in the release period. This is in line with the referee's prediction that "the radiation leakage compensation by amplification that artificially increases the mode lifetime. Reciprocally, the energy accumulation takes longer as well." In the new submission, we have incorporated the above discussion on lines 131-135. R5 "The larger the total scattering cross section, the longer the time required to accumulate sufficient energy to reach a stable state. In a similar vein, the accumulated energy is stored for a longer period after the incident wave is cut off. For the gain system, the radiation leakage is compensated by the gain metasurfaces in the release process that artificially increases the mode lifetime."

Referee #2 --Comment 2:
The manuscript reports about 40 time the scattering cross section in the active regime. It would be interesting to show how this number scale as a function of relevant parameter such as gain/loss ratio.

Authors Response:
This is a good point. For our single-channel superscatterer, an ultrathin gain metasurface is used to offer a complex surface impedance . The real part of determines the lossy or gain property, and its imaginary part determines the capacitive or inductive reactance. Using the simulated annealing algorithm, we design a single-channel superscatterer with the normalized scattering cross section of 44.5 at 3.733 GHz; the specific parameters can be found in the main text.
Based on this superscatterer structure, we change the gain/loss state to check the scattering cross section (Fig.   R4). In a lossy system with ( ) > 0, the scattering strength is inversely proportional to the loss. However, in a gain system with ( ) < 0, the scattering strength could increase remarkably. In principle, an infinitely large scattering cross section could be anticipated. In the new submission, we have incorporated the above discussion in the supplementary materials and noted it on lines 154-156.
"The increase/decrease of